which give rise to the GUT scale Yukawa ratios \(y_\tau /y_b = 3/2\)  and \(y_\mu /y_s = 9/2\)  [1]}.Alternatively, instead of \({\cal O}_3\) , one could also use a 5-plet messenger field, which then leads to a predicted ratio of \(y_\tau /y_b = - 3/2\) . For the phenomenology, the minus sign of course makes no difference. In these operators, the representations \(X\)  given as index for the brackets \((.)_X\)  specify the contractions of the GUT representations' indices inside the bracket.
We thus arrive at the following leading order structure of the GUT scale Yukawa matrices \(\textbf {Y}_e\)  and \(\textbf {Y}_d\) :
\(\textbf {Y}_e=\begin{pmatrix}0 && 0 && 0 \\0 && \frac{9}{2} && 0 \\0 && 0 && \frac{3}{2}\end{pmatrix}\cdot \textbf {Y}_d^T \:,\) 
